Fluid flow due to collective non-reciprocal motion of symmetrically-beatingartificial ciliaS. N. Khaderi, J. M. J. den Toonder, and P. R. Onck Citation: Biomicrofluidics 6, 014106 (2012); doi: 10.1063/1.3676068 View online: http://dx.doi.org/10.1063/1.3676068 View Table of Contents: http://bmf.aip.org/resource/1/BIOMGB/v6/i1 Published by the American Institute of Physics. Related ArticlesAsymmetry of red blood cell motions in a microchannel with a diverging and converging bifurcation Biomicrofluidics 5, 044120 (2011) Growth propagation of yeast in linear arrays of microfluidic chambers over many generations Biomicrofluidics 5, 044118 (2011) Making human enamel and dentin surfaces superwetting for enhanced adhesion Appl. Phys. Lett. 99, 193703 (2011) Synchronization of period-doubling oscillations in vascular coupled nephrons Chaos 21, 033128 (2011) The effects of inhomogeneous boundary dilution on the coating flow of an anti-HIV microbicide vehicle Phys. Fluids 23, 093101 (2011) Additional information on BiomicrofluidicsJournal Homepage: http://bmf.aip.org/ Journal Information: http://bmf.aip.org/about/about_the_journal Top downloads: http://bmf.aip.org/features/most_downloaded Information for Authors: http://bmf.aip.org/authors

Fluid flow due to collective non-reciprocal motion ofsymmetrically-beating artificial cilia

S. N. Khaderi,1,a) J. M. J. den Toonder,2 and P. R. Onck1,b)

1Zernike Institute for Advanced Materials, University of Groningen, Groningen, TheNetherlands2Eindhoven University of Technology, Eindhoven, The Netherlands

(Received 19 July 2011; accepted 16 December 2011; published online 20 January 2012)

Using a magneto-mechanical solid-fluid numerical model for permanently magnetic

artificial cilia, we show that the metachronal motion of symmetrically beating cilia

establishes a net pressure gradient in the direction of the metachronal wave, which

creates a unidirectional flow. The flow generated is characterised as a function of the

cilia spacing, the length of the metachronal wave, and a dimensionless parameter

that characterises the relative importance of the viscous forces over the elastic forces

in the cilia. VC 2012 American Institute of Physics. [doi:10.1063/1.3676068]

I. INTRODUCTION

In lab-on-a-chip devices, working fluids have to be pumped between micro-reaction cham-

bers through micron-sized channels. At these small length scales, the viscous forces dominate

over the inertial forces. Under this condition, a mechanical actuator has to move in a non-

reciprocal manner to cause a net fluid transport.21 In nature, micron-scale fluid manipulation is

often performed using periodically beating hair-like structures called cilia. An example of natu-

ral fluid manipulation systems is the expulsion of mucus from the lungs caused by the beating

of the cilia attached to the inner layer of mammalian trachea. The ciliary beat consists of dis-

tinct effective and recovery strokes, which leads to a non-reciprocal motion (a reciprocal

motion of actuator is one in which the forward motion is the same as backward motion). In

addition to the non-reciprocal motion of individual cilia, adjacent cilia beat with a constant

phase difference leading to a coordinated wave-like motion, which is referred to as metachronal

motion. Another example of fluid manipulation is the swimming of Cyanobacteria. Points on

the surface of Cyanobacteria oscillate symmetrically and generate waves of lateral displacement

along their surface. This wave-like motion causes fluid transport in one direction and the bacte-

ria swim in the opposite direction.7

Many examples have appeared in the recent literature of artificial cilia that mimic the natu-

ral ciliary motion using different physical actuation forces, imposed by electric fields, magnetic

fields or through base excitation.6,8–10,16,18,20,22,26 In most of the cases the actuation field is uni-

form,9,16,18,28 so that all artificial cilia beat in-phase, thus only focusing on the non-reciprocal

motion of individual cilia. The flow generated by synchronously beating cilia has been analysed

in terms of the dimensionless parameters that govern the cilia behaviour.1,2,10,11,16 The forma-

tion of the metachronal waves has been investigated using computational models, which suggest

that the coordinated motion is due to the hydrodynamic interaction between adjacent cilia, and

that the energy spent per cilium decreases in the presence of metachronal waves.12,13,18,27

Recently, it has been shown that the flow generated by magnetically driven non-reciprocally

beating artificial cilia is substantially enhanced and becomes unidirectional when the cilia beat

out-of-phase compared to synchronously beating cilia.10,17 By modelling cilia that beat out-of-

phase and possess only orientational asymmetry15 as a porous sublayer, it has been shown that

a)Presently at the Institute of High Performance Computing, Singapore.b)Electronic mail: p.r.onck@rug.nl.

1932-1058/2012/6(1)/014106/14/$30.00 VC 2012 American Institute of Physics6, 014106-1

BIOMICROFLUIDICS 6, 014106 (2012)

flow can be created in the direction of the metachronal wave.14 However, from a manufacturing

and implementation point-of-view, it is not straightforward to generate non-reciprocal motion of

the individual cilia. Therefore, it is of interest to investigate whether cilia can create a flow in

the absence of any asymmetry but in the sole presence of waves of lateral displacement, similar

to that of Cyanobacteria. This is the subject of the present article.

We study an array of permanently magnetic artificial cilia subjected to a non-uniform mag-

netic field that travels in space and time, such that the cilia beat symmetrically but out-of-phase.

The ciliary motion generates a unidirectional fluid flow in a direction opposite to the metachro-

nal wave. The fundamental mechanism which is responsible for the flow is investigated, and

the flow is quantified as a function of the parameters of the metachronal wave, cilia spacing,

and a physical dimensionless parameter that quantifies the relative importance of the viscous

forces compared to the elastic forces. We find that the flow reaches a maximum for critical val-

ues of the wavelength of the metachronal wave, which depends on the competition between the

elastic and viscous forces.

The article is organised as follows. The boundary value problem, governing equations,

solution approach, and parameters involved in the system are explained in Sec. II. The collec-

tive non-reciprocal motion is analysed in Sec. III A. The mechanisms that cause the flow are

analysed from Eulerian and Lagrangian points-of-view in Secs. III B1 and III B2, respectively.

The dependence of the flow on the system parameters is analysed in Sec. III C. We finally sum-

marise the results of the analysis in Sec. IV.

II. PROBLEM DEFINITION

We analyse the flow in an infinitely long channel of height H containing an array of two-

dimensional cilia of length L, thickness h, and inter-cilia spacing a, see Fig. 1. The cilia have

an elastic modulus E and possess a remanent magnetization of magnitude Mr oriented along

their length from the fixed end to the free end. A magnetic field is applied which is uniform

over each cilium but with a phase difference between adjacent cilia. The magnetic field experi-

enced by the ith cilium is

Bxi ¼ B0x sin xt� /ið Þ;Byi ¼ B0y; (1)

where x ¼ 2p=tref is the angular velocity, tref is the time period of oscillation, and the phase

/i ¼ 2pði� 1Þ=n. The phase difference between adjacent cilia is D/ ¼ 2p=n. The chosen form

of the phase enables the magnetic field to be periodic every nth cilium, so that the length of the

period or the wavelength of the applied magnetic field is k ¼ na. The applied magnetic field

FIG. 1. Schematic picture of the problem analysed. We study an infinitely long channel of height H containing equally

spaced cilia that are arranged perpendicular to the channel. A magnetic field which varies in space and time is used for

actuation. The resulting flow is opposite to the direction of the metachronal wave.

014106-2 Khaderi, den Toonder, and Onck Biomicrofluidics 6, 014106 (2012)

has a wave velocity k=tref to the right. A similar out-of-phase magnetic field has also been used

in a theoretical setting by Gauger et al.10 Experimentally, however, it is not straightforward to

create a magnetic field with a phase-lag between neighbouring cilia. It could be realised in an

approximate sense, for instance, by using a combination of an external Helmholtz coil (to create

Byi) and by locally actuating the cilia using short solenoids that “wrap-around” the microfluidic

channel (to create Bxi). By controlling the current in the solenoids, an out-of-phase magnetic

field can be generated that closely resembles the field used for the simulations (see the supple-

mentary material29). The magnetic body couple Nz acting on the cilia can be written as

Nz ¼ MrB0x. The velocity of the magnetic couple wave, which causes the out-of-phase motion

of the cilia, is also equal to k=tref . However, as there are a finite number of cilia per wave-

length, the metachronal wave velocity also depends on the cilia spacing a. The metachronal

velocity in cilia per seconds can be written as k=atref for 0 < a < k=2 and

�ðk=atrefÞ=ðk=a� 1Þ for k=2 < a < k, see Khaderi et al.17 The fluid to be propelled is assumed

to be incompressible and Newtonian with a viscosity l. The cilia are vertically straight when

no magnetic field is applied.

The dimensionless geometric parameter a=k ¼ D/=2p characterises the response of the

system together with H=L and a=L. The physical response of the system can be captured

through three dimensionless numbers: the fluid number Fn ¼ 12lL3=Eh3tref —the ratio of the

viscous forces to the elastic forces, the inertia number In ¼ 12qL4=Eh2t2ref —the ratio of cilia

inertia forces to the elastic forces and the magnetic number Mn ¼ 12B0xMrL2=l0Eh2 —the ratio

of magnetic forces to the elastic forces, where E and q are the elastic modulus and density of

the cilia.16 In this paper, we explore the effect of a=k, a=L, and Fn for a given value of In, Mn,

and H=L in the limit of low Reynolds numbers.

A. Governing equations

We now briefly discuss the coupled solid-fluid magneto-mechanical numerical model used

to study fluid propulsion by means of magnetically actuated artificial cilia.

1. Solid dynamic model

We model the cilia as elastic Euler-Bernoulli beams taking into consideration geometric

non-linearity in an updated Lagrangian framework. As a starting point for the Euler-Bernoulli

beam element formulation, we use the principle of virtual work19 and equate the virtual work

of the external forces at time tþ Dt (dWtþDtext ) to the internal work (dWtþDt

int ). The internal virtual

work is given by

dWtþDtint ¼

ðV

rd�þ qð€uduþ €vdvÞð Þ dV; (2)

where u and v are the axial and transverse displacements of a point on the beam and q is the

density of the beam. Furthermore, r is the axial stress and � is the corresponding strain, given

by

� ¼ @u

@xþ 1

2

@v

@x

� �2

�y@2v

@x2:

The external virtual work is

dWtþDtext ¼

ðfxduþ fydvþ Nz

@dv

@x

� �Adxþ

ðtxduþ tydv� �

bdx; (3)

where fx and fy are the magnetic body forces in the axial and transverse directions, Nz is the

magnetic body couple in the out-of-plane direction, tx and ty are the surface tractions and b is

the out-of-plane thickness of the cilia.

014106-3 Artificial cilia Biomicrofluidics 6, 014106 (2012)

We follow the approach used by Annabattula et al.3 to linearise and discretise the principal

of virtual work to get,

dpT KDpþM p:: tþDt � FtþDt

ext þ Ftint

� �¼ 0; (4)

where K is the stiffness matrix that combines both material and geometric contributions, M is

the mass matrix,5 FtþDtext is the external force vector, Ft

int is the internal force vector, Dp is the

nodal displacement increment vector, and p::

is the nodal acceleration vector. The nodal acceler-

ation vector is discretized in time using Newmark’s algorithm (using Newmark’s parameters

c ¼ 1:0 and b ¼ 0:5) so that Eq. (4) can be written in terms of the velocity of the beam. The

complete discretized equations of motion for the solid mechanics model can be found

elsewhere.16

2. Magnetostatics

To find the resulting magnetic forces, the magnetization of the cilia has to be calculated by

solving the Maxwell’s equations in the deformed configuration at every time increment. The

Maxwell’s equations for the magnetostatic problem with no external currents are

r � B ¼ 0 r�H ¼ 0; (5)

with the constitutive relation B ¼ l0ðM þHÞ; where B is the magnetic flux density (or mag-

netic induction), H is the magnetic field, M is the magnetization, and l0 is the permeability of

vacuum. Equation (5) is solved for M and B using the boundary element method.16 The mag-

netic couple per unit volume is given by N ¼ M � B0, where B0 has components Bxi and Byi.

As the simulations are two dimensional, the only non-zero component of magnetic body couple

is Nz which is the source for the external virtual work in Eq. (3). Since the applied magnetic

field is uniform for each cilium, the magnetic body forces due to field gradients are absent.

3. Fluid dynamics and solid fluid coupling

We study the flow created by artificial cilia in the limit of low Reynolds number. The fluid

is assumed to be Newtonian and incompressible. The physical behaviour of the fluid is gov-

erned by the Stokes equation,

�rpþ 2lr � D ¼ 0;r � u ¼ 0;

(6)

where p is the pressure in the fluid, D is the rate of deformation tensor, u is the velocity of the

fluid, and l is the viscosity of the fluid. The set of equations in Eq. (6) is solved using Eulerian

finite elements based on the Galerkin method. The fluid domain is discretized into quadrilaterals

in which the velocity and pressure of the fluid are interpolated quadratically and linearly,

respectively. The velocity is calculated at the vertices, mid-sides, and mid-point of the quadri-

lateral, and the pressure is calculated at the vertices. The solid and fluid domains are coupled

by imposing the constraint that the velocity at the nodes of the solid beam is equal to the veloc-

ity of the surrounding fluid (point collocation method). This coupling is established with the

help of Lagrange multipliers using the fictitious domain method. Details of the Eulerian finite

element model and the coupling procedure can be found in van Loon et al.25

To perform the numerical simulations, we choose a unit-cell whose width is equal to one

wavelength containing k=a cilia. The left and right end of the unit-cell are periodic in velocity,

while the top and bottom boundaries are no-slip boundaries. We perform simulations for vari-

ous values of the phase difference a=k, while the inter-cilia spacing a=L is maintained constant.

Therefore, as k is increased the number of cilia in a unit-cell also increases. The top and bottom

of the unit-cell are the channel walls, on which no-slip boundary conditions are applied,

014106-4 Khaderi, den Toonder, and Onck Biomicrofluidics 6, 014106 (2012)

utop ¼ ubottom ¼ 0;

while the left and right ends are periodic in velocity

uleft ¼ uright:

4. Solution procedure

The solution procedure is as follows. The Maxwell’s equations are solved at every time

instant to solve for the magnetic field. From the magnetic field, the magnetic body couple act-

ing on the cilia is calculated and is provided as an external load to the coupled solid-fluid

model, which simultaneously solves for the cilia velocity, and the velocity and pressure of the

fluid. The velocity of the cilia is integrated using Newmark’s algorithm to obtain its new posi-

tion, and the procedure is repeated. The particles and streamlines are obtained from the velocity

field in the fluid using the visualization software Tecplot.24 Also, here care should be taken to

accurately resolve the velocity field.

III. RESULTS

A. Collective non-reciprocal motion

We first show that the out-of-phase motion of the cilia leads to a collective non-reciprocal

motion. To elucidate this, we plot the schematic positions of the cilia at time instances tref=4

after and before the first cilium has reached its extreme position (at time te) in Fig. 2. Note that

the cilia will deform by bending—when a magnetic body couple Nz is applied—and will not

remain straight as is assumed in this section for simplicity. The thin and thick lines represent

the extreme and current positions of the cilia, respectively. The arrows represent the direction

of motion of the cilia. We would like to remind that for non-reciprocal motion of an actuator,

its position after and before an extreme position is reached should not be identical. In the cases

of cilia beating in synchrony and anti-phase, the positions of all the cilia before (te � tref=4)

and after the extreme position (te þ tref=4) is the same; hence the motion is reciprocal. How-

ever, in the case for cilia moving out-of-phase, even though the positions of the cilia 1 and 3

are identical before and after te, the positions of cilia 2 and 4 are not. Therefore, even though

every cilium performs a reciprocal motion, the cilia collectively perform a non-reciprocal

motion and will cause a net fluid flow in microchannels. The non-reciprocal motion due to the

out-of-phase motion may be quantified by the parameter a=k. The collective motion of the cilia

is reciprocal for ak ¼ m=2 (where m is an integer), otherwise the collective motion is

non-reciprocal. The fluid flow due to the non-reciprocal motion is quantitatively analysed in the

subsequent sections.

B. The fundamental mechanism

1. Eulerian point-of-view

To illustrate the fundamental mechanism that creates the flow, we take a=L ¼ 2=7,

a=k ¼ 1=7, H=L ¼ 4, Fn ¼ 0:15, In ¼ 3� 10�3, and Mn ¼ 2:25 (based on Bx0). As a=k ¼ 1=7,

the metachronal wave travels to the right with a speed k=tref . Figure 3(a) shows the pressure

contours with the streamlines superimposed and Fig. 3(b) shows the contours of the absolute

value of the horizontal component of the velocity. For clarity, the results are shown for two

unit-cells. Because of the travelling magnetic wave and periodicity of the system, the pressure

and velocity profiles in the channel remain unchanged in time, but travel with a constant veloc-

ity in the direction of the applied magnetic wave (to the right).

At the instant depicted, cilia 2, 3, 4 and 9, 10, 11 move to the right, while cilia 6, 7 and

13, 14 move to the left. The other cilia are nearly stationary (zero velocity). Due to the instan-

taneous velocity of the cilia, high pressure (hp) and low pressure (lp) regions develop (red and

blue regions in Fig. 3(a)). Fluid is squeezed out from the hp region and sucked in by the lp

014106-5 Artificial cilia Biomicrofluidics 6, 014106 (2012)

FIG. 2. Schematic positions of the cilia at instances tref=4 after (te þ tref=4) and before (te � tref=4) the extreme position

has been reached by the first cilium at te. The time te corresponds to the extreme position of the first cilium from the left.

The arrows represent the direction of motion of the cilia. In the cases of cilia beating synchronously and anti-phase, the

positions of all the cilia before and after the extreme position is the same; hence the motion is reciprocal. However, it is not

the case for cilia moving out-of-phase; hence the motion is non-reciprocal.

014106-6 Khaderi, den Toonder, and Onck Biomicrofluidics 6, 014106 (2012)

regions, as a result of which a series of counter-rotating vortices are formed in the channel.

Since the distance between the hp and lp regions opposite to the wave direction is smaller, the

pressure gradient is larger, so that the counter-clockwise vortices are stronger (see Fig. 3(b)).

As a result, the velocity distribution has a dominant horizontal component to the left. Integrat-

ing the velocity profile over the channel height results in a net flux to the left. Conservation of

mass dictates that the flux at every vertical section through the channel is the same. Since the

entire periodic profile, as depicted in Fig. 3, travels continuously to the right at a constant pace,

the flux remains constant in time. Clearly, the flux magnitude and direction can be directly

deduced from the instantaneous pressure distribution profile of Fig. 3 as analysed in the

following.

Fluid flow occurs in the direction opposite to the net pressure gradient. This pressure

gradient is governed by the magnitude of the pressure in the lp and hp regions and the distance

between them. The former is governed by the velocity of the individual cilia, whereas the latter

is determined by the deflection d of the cilia tip. Since the velocity and displacements of the

cilia are controlled by the magnetic field and its rate of change, it can be deduced that for aconsiderably smaller than k the net pressure gradient scales with lxd2=k3 (see Appendix). As

this pressure gradient is positive, the flow occurs in the negative x-direction. Thus, the flow

direction is opposite to the metachronal wave, and scales with the square of the amplitude of

deflection. When the direction of the applied magnetic wave is reversed, the pressure profile,

which is dictated by the cilia velocity, remains alternating. However, the deformed configura-

tion of the cilia changes in such a way that the net pressure gradient is now negative; this

creates a flow to the right (again opposite to the metachronal wave). Fluid flow created by

oscillating cilia whose motion is kinematically prescribed has been analysed recently using a

continuum approach.14 The formation of the vortices was also observed in this work. A rigor-

ous mathematical analysis of the fluid flow (following similar scaling arguments as Taylor23)

induced by a longitudinally oscillating sheet whose material particles comprise of the tip of the

cilia also predicts that the flow will occur in the direction opposite to the metachronal wave.4

FIG. 3. Fundamental mechanism causing fluid flow: (a) Contours of pressure and (b) Contours of absolute velocity in

x-direction for a=L ¼ 2=7 and a=k ¼ 1=7 (wave moving to the right) at t ¼ 0:35tref . Due to the velocity of cilia, regions of

positive and negative pressure are established in the channel. The deformed position of the cilia causes a lower pressure

gradient in direction of the wave compared to that of in the opposite direction. This leads to a high velocity and a net flow

to the left.

014106-7 Artificial cilia Biomicrofluidics 6, 014106 (2012)

2. Lagrangian point-of-view

The previous analysis focussed on the channel flow from an Eulerian point-of-view.

Instead, we can also adopt a Lagrangian view point and track the motion of fluid particles in

time. In the following, all the parameters of the Sec. III B 1 are kept constant, except for H,

which we now choose to be 2L. Figure 4 shows a portion of the region between the tips of the

cilia and the top boundary. We follow the motion of the fluid particles (that initially form a

straight vertical line) when they come under the influence of the travelling vortices (see Fig. 4).

The contours represent the absolute velocity in the x-direction, and the direction of velocity is

represented by the streamlines. From the velocity field, we can see that the out-of-phase motion

of cilia creates a series of counter-acting vortices, and that the velocity field travels to the right,

which is also the direction of propagation of the applied magnetic field wave.

We focus our attention on the second particle from the bottom. At t ¼ 0 the particle is

between two vortices. The velocity of the particle is such that it moves downwards. As time

progresses, at t ¼ 0:2tref the position of the particle is such that it has a low velocity to the right

due to the presence of the clockwise vortex. At t ¼ 0:4tref , the particle moves away from the

influence of the clockwise vortex, towards the counter-clockwise vortex. Now the particle has a

velocity such that it moves upwards. At t ¼ 0:6tref , when the particle is under the influence of

the stronger counter-clockwise vortex, it has a higher velocity compared to the instance when

the particle was under the influence of less strong clockwise vortex (compare Figs. 4(b) and

4(c)). Therefore, the particle effectively moves to the left. The displacement perpendicular to

the channel is less for the particles near the top boundary. This leads to equal velocities in the

channel direction when they come under the influence of the clockwise and counterclockwise

vortices, which results in no net displacement of the particles near the top boundary.

It can be seen that the fluid particle near the free end of the cilia moves unidirectionally

and that its displacement is much larger compared to the displacement of the rest of the par-

ticles. Therefore, the contribution to the flow from the fluid particles near the cilia is much

larger compared to other particles. This results in a flow that is nearly unidirectional, even

though the cilia motion is oscillatory. An unidirectional flow is also observed when the cilia

motion is non-reciprocal in the presence of metachronal waves and inertia.15 A similar approach

(Lagrangian point-of-view) was used by Ehlers et al.7 to analyse the swimming of microorgan-

isms (based on tangential travelling waves).

C. Parametric study

We now characterise the flow generated by the cilia as a function of the system parameters

such as the cilia spacing a, wavelength k, amplitude of cilia deflection, and fluid number Fn.

The output of the cilia is quantified by the area flow created per cycle, which is found by inte-

grating the instantaneous velocity flux through the channel over a representative cycle. The

nature of the fluid flow, as seen in Sec. III B 2, is unidirectional. We focus our attention on the

cases where the direction of the metachronal wave velocity is positive, i.e., a < k=2, and per-

form the analysis for H ¼ 2L, Mn ¼ 1:13, Fn ¼ 0:15, and In ¼ 3� 10�3 unless mentioned

otherwise.

Figure 5(a) shows the flow and the metachronal wave velocity as a function of a=k. When

a� k, the cilia move in-phase (not shown). When a ¼ k=2, the positive wave velocity and the

negative wave velocity are equal and this leads to a standing wave (see Khaderi et al.17) that

causes the cilia motion to be reciprocal (see Fig. 2(b)). Under these conditions no fluid flow is

observed. As the magnetic wave travels to the right (see Fig. 1), the fluid flows to the left; the

opposite occurs for k=2 > a > k. To get more insight into the behaviour of the system, we plot

the flow as a function of L=k for the cases shown in Figs. 5(a) and 5(b). The arrow shows the

direction of increasing k. The fluid flow increases when the inter-cilia spacing is decreased, and

for a given inter-cilia spacing, the flow initially increases with k, reaches a maximum at k ¼ 4Land then decreases.

As shown in Appendix the pressure gradient responsible for the flow increases when the

wavelength decreases, as well as when the deflection of the cilia increases. However, there is

014106-8 Khaderi, den Toonder, and Onck Biomicrofluidics 6, 014106 (2012)

an additional dependence. When the wavelength of the magnetic field is increased in Fig. 5(b)

for a particular cilia spacing, the pressure between the cilia decreases, which reduces the hydro-

dynamic drag and causes an increased deformation of the cilia (see Fig. 6). On the other hand,

the pressure gradient will decrease with an increase in the wavelength,30 so that the flow cre-

ated is due to the relative competition between these two effects. The deflection effect domi-

nates when the wavelength is small, creating an increased flow as we increase the wavelength

until k ¼ 4L. When we increase the wavelength any further, the effect of the increasing

FIG. 4. Motion of particles with time: The field-of-view is the region between the oscillating end of cilia and top boundary

of the unit-cell with the bottom left corner at ðx; yÞ ¼ ð0:5L; 0:95LÞ and the top right corner at ðx; yÞ ¼ ð1:55L; 2LÞ. The

velocity in the channel direction is larger in the direction opposite to the wave than in the direction of the wave. The white

curves represent the trajectory of particles and the black dots represent the particles. Particles near the cilia move unidirec-

tionally and show larger displacement, whereas the particles near the top boundary do not show any displacement. An ani-

mation of the motion of the fluid particles is added as a supporting information (enhanced online) [URL: http://dx.doi.org/

10.1063/1.3676068.1].

014106-9 Artificial cilia Biomicrofluidics 6, 014106 (2012)

deflection is overcome by the decreasing pressure gradient, which causes the flow to decrease.

This is in contrast to the situation where the cilia sweep an asymmetric area (see Khaderi

et al.17), where it is shown that the flow created does not depend on the phase difference

between adjacent cilia. This is due to the different mechanisms that drive the flow: The pressure

gradient in the current situation and the asymmetric area in the work of Khaderi et al.17

To investigate the possible contribution of shape asymmetry to the flow generated, we have

calculated the area swept by the tip of the cilia. For all the cases reported the normalized swept

area (area swept/ðpL2=2Þ) is on the order of 10�5, which can be estimated16 to create a flow

that is four orders of magnitude smaller than the flow reported in the current manuscript.

Clearly, the fluid transport generated by shape asymmetry is negligible.

The flow as a function of the fluid number (Fn) for various cilia spacings at L=k ¼ 1=4 is

shown in Fig. 7; the flow decreases with an increase in the fluid number. As the viscous forces

increase with the fluid number, the cilia exhibit lower deflections when Fn is increased (see the

inset in Fig. 7). As the pressure gradient scales with the cilia deflection, the flow created

decreases with the deflection.

FIG. 5. Flow as a function of (a) phase difference a=k and (b) inverse wavelength L=k for various cilia densities a=L. The

fluid flows in a direction opposite to the metachronal velocity (shown using dashed lines in (a)). The flow is maximum

when k ¼ 4L. The arrow shows the direction of increase of k.

FIG. 6. Maximum transverse displacement as a function of L=k for different cilia spacings (a=L). When L=k is large the

phase difference between adjacent cilia is large, causing them to move against each other, which leads to high drag forces

on the cilia causing low deformation.

014106-10 Khaderi, den Toonder, and Onck Biomicrofluidics 6, 014106 (2012)

FIG. 7. Flow as a function of the fluid number (Fn) for various cilia spacing at L=k ¼ 1=4. The maximum transverse dis-

placement of the free end is shown as an inset.

FIG. 8. Fluid transported as a function of L=k for different fluid numbers and cilia spacings.

TABLE I. Maximum transverse displacement (normalised with L) for the cases shown in Fig. 8. Also shown is the percent-

age increase in the displacements as the wavelength is increased.

a=L ¼ 2=3 a=L ¼ 4=5

a=k Fn ¼ 0:3 Fn ¼ 0:15 Fn ¼ 0:075 Fn ¼ 0:3 Fn ¼ 0:15 Fn ¼ 0:075

0.333 0.0345 0.0579 0.868 0.040 0.0672 0.0945

0.250 0.0426 0.0706 0.969 0.047 0.0775 0.1015

0.200 0.0498 0.0800 1.030 0.053 0.0843 0.1055

0.167 0.0556 0.0866 1.060 0.058 0.0886 0.1077

0.143 0.0599 0.0908 1.087 0.061 0.0912 0.1090

0.125 0.0632 0.0935 1.100 0.063 0.0957 0.1098

Percentage increase 83.1 61.4 26.7 56.6 42.4 16.2

014106-11 Artificial cilia Biomicrofluidics 6, 014106 (2012)

The flow as a function of L=k for different fluid numbers is shown in Fig. 8 for various cilia

spacings. The flow reaches a maximum at different values of L=k for different fluid numbers.

The maximum flow occurs at shorter wavelengths (i.e., at larger L=k values) as the fluid number

is reduced. The tip displacements corresponding to Fig. 8 are shown in Table I. The percentage

increase in the displacement decreases as the fluid number is decreased. When the wavelength is

increased, as mentioned earlier, the flow increases because of the increase of the displacement

and decreases due to the decreased pressure gradient. However, as the fluid number is decreased,

the increase of flow because of the increase of the displacement is limited. Therefore, the maxi-

mum fluid transport takes place at smaller wavelengths (i.e., at larger L=k values).

When k� a, the fluid experiences an oscillating surface whose material particles are made

up of the tip of cilia. The velocity of propulsion in such a case is given by the envelope theory

under the assumption that the cilia spacing is much smaller than the wavelength. Brennen4 has

shown that for a continuous distribution of cilia, the fluid velocity scales with the inverse

square of the wavelength, the frequency of oscillation x, and the square of the amplitude of os-

cillation (in accordance with the physical scaling laws derived earlier by Taylor23). Moreover,

the flow is in the direction opposite to the wave velocity. Our model also captures all these

aspects. In the following, we show that the flow scales with the square of the amplitude of os-

cillation even at large amplitudes of deflection and spacing of cilia (a � L).

To do so, we examine the fluid flow dependence on the magnitude of the transverse deflec-

tion of the cilia. We take the case of a ¼ 2L and a=k ¼ 1=3. The magnitude of displacement is

increased by increasing the magnetic number from 2.25 to 34. The flow as a function of dis-

placement is shown in Fig. 9. The flow has a quadratic dependence on the deflection until the

deflection is 40% of the cilia length.

IV. CONCLUSIONS

In this article, we analysed the fluid transport created by cilia that beat symmetrically and

out-of-phase when actuated by a non-uniform magnetic field, leading to the formation of meta-

chronal waves. Although at the scale of individual cilia the beating is reciprocal, because of the

metachronal waves the cilia collectively exhibit a non-reciprocal motion. Using a magneto-

mechanical solid-fluid numerical model we analysed the fundamental mechanisms that cause

this fluid flow. The out-of-phase motion of cilia creates a net pressure gradient, which results in

a unidirectional flow whose direction is opposite to the direction of the wave. The flow

increases with the tip deflection of cilia and decreases with the wavelength. Analysis of the

motion of fluid particles reveals that the major contribution to the fluid flow comes from the

particles located near the free end of the cilia. The flow created reaches a maximum value at

critical values of wavelengths (k=L), which decreases with the fluid number.

FIG. 9. Flow as a function of deflection.

014106-12 Khaderi, den Toonder, and Onck Biomicrofluidics 6, 014106 (2012)

APPENDIX: CALCULATION OF THE NET PRESSURE GRADIENT IN THE UNIT-CELL

In this appendix, we derive an expression for the net pressure gradient in the channel due tothe out-of-phase motion of cilia. For simplicity, we take the case where 0 < a < k; i.e., the meta-chronal wave velocity and the applied wave velocity are the same (to the right). The orientation ofthe cilia is dictated by the magnetic field vector, while their velocity is dictated by the rate ofchange of the magnetic field vector. The current x position of the tip of a cilium isx0 þ d0 sin xt� 2px0=kð Þ and its velocity is d0x cos xt� 2px0=kð Þ, where d0 is assumed to scalewith the amplitude of the applied magnetic field in the x-direction B0x and x0 ¼ ði� 1Þa is theposition of the cilia base. Figure 10 shows the current velocity (solid lines) and displacement(dashed lines) for a=L ¼ 2=7, with the wave travelling to the right. The pressure will be positivewhen two cilia come close and will be negative when they move apart, i.e., the positive and nega-tive pressures will occur when the velocity gradient is negative and positive, respectively. At anytime t, this happens at tk=T � k=4 (negative pressure), tk=T þ k=4 (positive pressure), andtk=T þ 3k=4 (negative pressure).

The position of cilia with its velocity at a particular time instance is shown in Fig. 10. It canbe seen that two neighbouring cilia move apart in regions 1 and 3, thereby generating a negativepressure �p. In region 2, two neighbouring cilia come closer, this generates a positive pressure p.(We have assumed that the positive pressure is equal to the negative pressure). At any time t, theposition of region 1 is

x1 ¼t

Tk� k

4þ d0 sin xt� 2p

kt

Tk� k

4

� �� �¼ t

Tk� k

4þ d0: (A1)

Similarly, the respective positions of the regions 2 and 3 are

x2 ¼t

Tkþ k

4� d0; (A2)

x3 ¼t

Tkþ 3

k4þ d0: (A3)

The pressure gradient between regions 1 and 2 can be written to be proportional to

p� ð�pÞx2 � x1

¼ 4p

k� 4d0

;

and the pressure gradient between the region 2 and 3 can be written to be proportional to

FIG. 10. Velocity (arrows and solid lines) and deformation (dashed line) of the cilia at any time instant, neglecting the

inter-cilium interaction.

014106-13 Artificial cilia Biomicrofluidics 6, 014106 (2012)

�p� ðpÞx3 � x2

¼ � 4p

kþ 4d0

:

The average pressure gradient in the unit-cell is now given by,

16pd0=ðk2 � 16d20Þ:

By invoking that p scales with ld0x=k, it can be seen that the pressure gradient that drives theflow increases with an increase of the cilia deflection and with a decrease of the wavelength,

16ld20x=kðk2 � 16d2

0Þ:

Assuming small cilia deflection (k >> d0), we observe that the average pressure gradient scales

with ðld20x=k

3Þ. Therefore, the pressure gradient is positive when the wave is moving in the posi-tive x-direction; this creates a flow in the negative x-direction. Although the applicability of theexpression for the pressure gradient is limited to large wavelengths and small cilia deflections, itexplains all the trends observed in the simulations (the direction of flow, the increase in flowbecause of decreasing wavelength and increasing cilia deflection).

1A. Alexeev, J. M. Yeomans, and A. C. Balazs, Langmuir 24, 12102 (2008).2A. Alexeev, J. M. Yeomans, and A. C. Balazs, Langmuir 24(21), 12102 (2008).3R. K. Annabattula, W. T. S. Huck, and P. R. Onck, J. Mech. Phys. Solids 58, 447 (2010).4C. Brennen, J. Fluid Mech. 65(04), 799 (1974).5Robert D. Cook, D. S. Malkus, M. E. Plesha, D. S. Malkus, and M. E. Plesha, Concepts and Applications of FiniteElement Analysis (Wiley, New York, 2001).

6Jaap den Toonder, Femke Bos, D. Broer, L. Filippini, M. Gillies, Judith de Goede, Titie Mol, Mireille Reijme, W. Talen,H. Wilderbeek, V. Khatavkar, and P. Anderson, Lab Chip 8(4), 533 (2008).

7K. M. Ehlers, A. D. Samuel, H. C. Berg, and R. Montgomery, Proc. Natl. Acad. Sci. USA 93(16), 8340 (1996).8B. A. Evans, A. R. Shields, R. Lloyd Carroll, S. Washburn, M. R. Falvo, and R. Superfine, Nano Letters 7(5), 1428(2007).

9F. Fahrni, M. W. J. Prins, and Leo J. van IJzendoorn, Lab Chip 9, 3413 (2009).10E. M. Gauger, M. T. Downton, and H. Stark, Eur. Phys. J. E 28, 231 (2009).11R. Ghosh, G. A. Buxton, O. Berk Usta, Anna C. Balazs, and Alexander Alexeev, Langmuir 26, 2963 (2010).12S. Gueron and K. Levit-Gurevich, Proc. Natl. Acad. Sci. USA 96(22), 12240 (1999).13S. Gueron, K. Levit-Gurevich, N. Liron, and J. J. Blum, Proc. Natl. Acad. Sci. USA 94(12), 6001 (1997).14J. Hussong, W. P. Breugem, and J. Westerweel, J. Fluid Mech. 684, 137 (2011).15S. N. Khaderi, M. G. H. M. Baltussen, P. D. Anderson, J. M. J. den Toonder, and P. R. Onck, Phys. Rev. E 82, 027302

(2010).16S. N. Khaderi, M. G. H. M. Baltussen, P. D. Anderson, D. Ioan, J. M. J. den Toonder, and P. R. Onck, Phys. Rev. E

79(4), 046304 (2009).17S. N. Khaderi, J. M. J. den Toonder, and P. R. Onck, J. Fluid Mech. 688, 44 (2011).18Y. W. Kim and R. R. Netz, Phys. Rev. Lett. 96(15), 158101 (2006).19L. E. Malvern, Introduction to the Mechanics of a Continuous Medium (Prentice-Hall, 1977).20Kieseok Oh, Jae-Hyun Chung, S. Devasia, and J. J. Riley, Lab Chip 9(11), 1561 (2009).21E. M. Purcell, Am. J. Phys. 45(1), 3 (1977).22A. R. Shields, B. L. Fiser, B. A. Evans, M. R. Falvo, S. Washburn, and R. Superfine, Proc. Natl. Acad. Sci. 107(36),

15670 (2010).23G. Taylor, Proc. R. Soc. London A 209, 447 (1951).24Tecplot, Tec360 User Manual (2008).25R. van Loon, P. D. Anderson, and F. N. van de Vosse, J. Comput. Phys. 217, 806 (2006).26Casper L. van Oosten, Cees W. M. Bastiaansen, and Dirk J. Broer, Nature Mater. 8, 677 (2009).27A. Vilfan and F. Julicher, Phys. Rev. Lett. 96(5), 058102 (2006).28M. Vilfan, A. Potocnik, B. Kavcic, N. Osterman, I. Poberaj, A. Vilfan, and D. Babic, Proc. Natl. Acad. Sci. 107, 1844

(2010).29See supplementary material at http://dx.doi.org/10.1063/1.3676068 for the simulations of the magnetic field caused by

the solenoids.30Note that the pressure gradient will continue to decrease with an increase in wavelength, whereas its increase because of

the increased cilia deflection is limited.

014106-14 Khaderi, den Toonder, and Onck Biomicrofluidics 6, 014106 (2012)